Simplify and expand the following expression: $ \dfrac{r}{2r + 9}+\dfrac{r - 10}{5r + 3} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2r + 9)(5r + 3)$ Multiply the first term by $\dfrac{5r + 3}{5r + 3}$ $ \begin{align*} \dfrac{r}{2r + 9} \times \dfrac{5r + 3}{5r + 3} & = \dfrac{(r)(5r + 3)}{(2r + 9)(5r + 3)} \\ & = \dfrac{5r^2 + 3r}{(2r + 9)(5r + 3)}\end{align*} $ Multiply the second term by $\dfrac{2r + 9}{2r + 9}$ $ \begin{align*} \dfrac{r - 10}{5r + 3} \times \dfrac{2r + 9}{2r + 9} & = \dfrac{(r - 10)(2r + 9)}{(5r + 3)(2r + 9)} \\ & = \dfrac{2r^2 - 11r - 90}{(5r + 3)(2r + 9)}\end{align*} $ Now we have: $ = \dfrac{5r^2 + 3r}{(2r + 9)(5r + 3)} + \dfrac{2r^2 - 11r - 90}{(5r + 3)(2r + 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5r^2 + 3r + 2r^2 - 11r - 90}{(2r + 9)(5r + 3)} $ $ = \dfrac{7r^2 - 8r - 90}{(2r + 9)(5r + 3)}$ Expand the denominator: $ = \dfrac{7r^2 - 8r - 90}{10r^2 + 51r + 27}$